I. ALGEBRA 



1.00 Algebraic Identities. 



1. a n - b n = (a - b)(a n ~ l + a n ~ 2 b + a n ~*b 2 + ..... + ab n ~ 2 + 



2. a n b n = (a + b) (a n ~ l - a n ~ 2 b + a n ~*b 2 - ..... =F ab n ~ 2 



n odd: upper sign. 

 n even: lower sign. 



3. (x + ai)(x + (h) . . . . . (* + an) = x n + Pix n ~ l + P2X n ~ z + . . . . 



+ P n -!X + P n . 



P l = 0i + 2 + ...... + fln- 



Pfc = sum of all the products of the a's taken k at a time. 



P n = 



4. (a 2 + 6 2 )(a 2 + /3 2 ) = (aa =F bfi) 2 + (a/3 ba) 2 . 



5. (a 2 - 6 2 )(a 2 - P 2 ) = (aa 6/3) 2 - (a/3 6a) 2 . 



6. (a 2 + b 2 + c 2 )(a 2 + /3 2 + 7 2 ) = (<*a + 6]3 + cy) 2 + (67 - /3c) 2 + (ca - 



+ (aft - ab) 2 . 



7 . ( a 2 + ^2 + C 2 + ^2)( a 2 + ^2 + T 2 + 52) = ( aa + ^ + CT + ^5)2 



+ 0/3 - 6a + c5 - dy) 2 + (07 - bd - ca + dj3) 2 + (ad + by - eft - da) 



8. (ac - bd) 2 + (ad + be) 2 = (ac + bd) 2 + (ad - be) 2 . 



9. (a + 6) (6 + c)(c + a) = (a + b + c)(ab + be + ca)-abc. 



10. (a + b)(b + c)(c + a) = a 2 (6 + c) + ^(c + a) + c 2 (o + 6) + ^abc. 



n. (a + b)(b + c)(c + a) = fc(> + c) + ca(c + a) + ab(a + b) + 2abc. 



12. 3(0 + b)(b + c)(c + a) = (a + b + c)*- O 3 + ^ 3 + c 8 )- 



13. (b - a)(c - a)(c -b) = a 2 (c - b) + b 2 (a - c) + c 2 (b - a). 



14. (b -a)(c- a)(c - b) = a(b 2 - c 2 ) + b(c 2 - a 2 ) + c(a 2 - b 2 ). 



15. (b - a)(c - a)(c - b) = bc(c - b) + ca(a - c) + ab(b - a). 



!6. - b) 2 +(b- c) 2 + (c - a) 2 = 2 [0 - b)(a - c) + (b - a)(b - c) 

 + (e-a)(e-b)l. 



17. a?(b 2 - c 2 ) + &V - a 2 ) + cV - b 2 ) = (a - b)(b - c)(a - c)(ab + be + ca). 



18. + b + c)(a 2 + b 2 + c 2 ) = bc(b + c) + ca(c + a) + a5(a + b) + a 3 + 6 s + c 



19. + 6 + c)(fa + ca + aft) = o 2 (6 + c) + b 2 (c + a) + c 2 (a + 6) 



20. (6 + c - a )(c + a - b)(a + b - c) = a 2 (b + c) + & 2 + <0 + ^ 2 



